A065463 -id:A065463 - cp's OEIS frontend

A065490 Exponents in expansion of constant A065463 as Product_{n>1} zeta(n)^(-a(n)).

0, 1, -1, 1, -2, 3, -4, 5, -8, 13, -18, 25, -40, 62, -90, 135, -210, 324, -492, 750, -1164, 1809, -2786, 4305, -6710, 10460, -16264, 25350, -39650, 62057, -97108, 152145, -238818, 375165, -589520, 927200, -1459960, 2300346, -3626200

Offset: 1

N. J. A. Sloane, Nov 19 2001

sign

The sequence 1,1,1,1,2,3,4,5,8,13,18,25,40,62,90,135,... appears in Lehrer-Segal on p. 285, in the following context: Let V=Sum_{k>=1} V_k be the graded vector space H_*(PC^oo)[1], which has Poincaré series [or Poincare series] p(t)=t/(1-t^2). This sequence gives the dimensions of the free graded Lie algebra L on V.

Inverse Euler transform of F(1-n) where F() is Fibonacci numbers A000045. - Michael Somos, Jul 21 2003

Alois P. Heinz, Table of n, a(n) for n = 1..2000
G. I. Lehrer, Some sequences arising at the interface of representation theory and homotopy theory
G. I. Lehrer and G. B. Segal, Homology stability for classical regular semisimple varieties, Math. Zeit., 236 (2001), 251-290.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
N. J. A. Sloane, Transforms

Cf. A065463.

a[n_] := DivisorSum[n, (-1)^#*MoebiusMu[n/#]*(Fibonacci[#+1] + Fibonacci[# -1]-1)&]/n; Array[a, 40] (* Jean-François Alcover, Dec 03 2015, adapted from PARI *)

a(n)=if(n<1,0,sumdiv(n,d,(-1)^d*moebius(n/d)*(fibonacci(d+1)+fibonacci(d-1)-1))/n)

a(n) = (1/n)*Sum_{d|n} (-1)^d*mu(n/d)*(Fibonacci(d-1)+Fibonacci(d+1)-1). - Vladeta Jovovic, May 03 2003

a(n) ~ (-1)^n * phi^n / n, where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 09 2019

More terms and formula from Christian G. Bower, Aug 23 2002

A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 6, 55, 14, 57, 58, 59, 30, 61, 62, 21, 2, 65, 66, 67, 34, 69, 70, 71, 6, 73, 74, 15, 38, 77, 78

Offset: 1

R. Muller, Mar 15 1996

Multiplicative with a(p^e) = p.
Product of the distinct prime factors of n.
a(k)=k for k=squarefree numbers A005117. - Lekraj Beedassy, Sep 05 2006
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), b*c = A019554(n) = "outer square root" of n, and a(n) = lcm(a(b),c). Unless n is biquadrateful (A046101), a(n) = lcm(b,c). [Edited by Jeppe Stig Nielsen, Oct 10 2021, and Andrey Zabolotskiy, Feb 12 2025]
a(n) = A128651(A129132(n-1) + 2) for n > 1. - Reinhard Zumkeller, Mar 30 2007
Also the least common multiple of the prime factors of n. - Peter Luschny, Mar 22 2011
The Mobius transform of the sequence generates the sequence of absolute values of A097945. - R. J. Mathar, Apr 04 2011
Appears to be the period length of k^n mod n. For example, n^12 mod 12 has period 6, repeating 1,4,9,4,1,0, so a(12)= 6. - Gary Detlefs, Apr 14 2013
a(n) differs from A014963(n) when n is a term of A024619. - Eric Desbiaux, Mar 24 2014
a(n) is also the smallest base (also termed radix) for which the representation of 1/n is of finite length.  For example a(12) = 6 and 1/12 in base 6 is 0.03, which is of finite length. - Lee A. Newberg, Jul 27 2016
a(n) is also the divisor k of n such that d(k) = 2^omega(n). a(n) is also the smallest divisor u of n such that n divides u^n. - Juri-Stepan Gerasimov, Apr 06 2017

			G.f. = x + 2*x^2 + 3*x^3 + 2*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 2*x^8 + 3*x^9 + ... - _Michael Somos_, Jul 15 2018

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Masum Billal, Divisible Sequence and its Characteristic Sequence, arXiv:1501.00609 [math.NT], 2015, theorem 11 page 5.
Henry Bottomley, Some Smarandache-type multiplicative sequences
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.
Neville Holmes, Integer Sequences [Broken link]
Serge Lang, Old and New Conjectured Diophantine Inequalities, Bull. Amer. Math. Soc., 23 (1990), 37-75. see p. 39.
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.
D. H. Lehmer, Euler constants for arithmetical progressions, Collection of articles in memory of Juriĭ Vladimirovič Linnik. Acta Arith. 27 (1975), 125--142. MR0369233 (51 #5468). See N_k on page 131.
Ivar Peterson, The Amazing ABC Conjecture
Paul Tarau, Emulating Primality with Multiset Representations of Natural Numbers, in Theoretical Aspects of Computing, ICTAC 2011, Lecture Notes in Computer Science, 2011, Volume 6916/2011, 218-238
Paul Tarau, Towards a generic view of primality through multiset decompositions of natural numbers, Theoretical Computer Science, Volume 537, 5 June 2014, Pages 105-124.
Wikipedia, Radical of an integer.

See A007913, A062953, A000188, A019554, A003557, A066503, A087207 for other properties related to square and squarefree divisors of n.
More general factorization-related properties, specific to n: A020639, A028234, A020500, A010051, A284318, A000005, A001221, A005361, A034444, A014963, A128651, A267116.
Range of values is A005117.
Cf. A048803, A019565, A024619, A053462, A060735, A073355 (partial sums), A079277, A097945, A129132, A225546.
Bisections: A099984, A099985.
Sequences about numbers that have the same squarefree kernel: A065642, array A284311 (A284457).
A003961, A059896 are used to express relationship between terms of this sequence.
Cf. A001615, A008683, A076479, A078615.

a007947 = product . a027748_row  -- Reinhard Zumkeller, Feb 27 2012

[ &*PrimeDivisors(n): n in [1..100] ]; // Klaus Brockhaus, Dec 04 2008

with(numtheory); A007947 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul(t1[i][1],i=1..nops(t1)); end;
A007947 := n -> ilcm(op(numtheory[factorset](n))):
seq(A007947(i),i=1..69); # Peter Luschny, Mar 22 2011
A:= n -> convert(numtheory:-factorset(n),`*`):
seq(A(n),n=1..100); # Robert Israel, Aug 10 2014
seq(NumberTheory:-Radical(n), n = 1..78); # Peter Luschny, Jul 20 2021

rad[n_] := Times @@ (First@# & /@ FactorInteger@ n); Array[rad, 78] (* Robert G. Wilson v, Aug 29 2012 *)
Table[Last[Select[Divisors[n],SquareFreeQ]],{n,100}] (* Harvey P. Dale, Jul 14 2014 *)
a[ n_] := If[ n < 1, 0, Sum[ EulerPhi[d] Abs @ MoebiusMu[d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 15 2018 *)
Table[Product[p, {p, Select[Divisors[n], PrimeQ]}], {n, 1, 100}] (* Vaclav Kotesovec, May 20 2020 *)

a(n) = factorback(factorint(n)[,1]); \\ Andrew Lelechenko, May 09 2014

for(n=1, 100, print1(direuler(p=2, n, (1 + p*X - X)/(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from sympy import primefactors, prod
def a(n): return 1 if n < 2 else prod(primefactors(n))
[a(n) for n in range(1, 51)]  # Indranil Ghosh, Apr 16 2017

def A007947(n): return mul(p for p in prime_divisors(n))
[A007947(n) for n in (1..60)] # Peter Luschny, Mar 07 2017

(define (A007947 n) (if (= 1 n) n (* (A020639 n) (A007947 (A028234 n))))) ;; ;; Needs also code from A020639 and A028234. - Antti Karttunen, Jun 18 2017

If n = Product_j (p_j^k_j) where p_j are distinct primes, then a(n) = Product_j (p_j).
a(n) = Product_{k=1..A001221(n)} A027748(n,k). - Reinhard Zumkeller, Aug 27 2011
Dirichlet g.f.: zeta(s)*Product_{primes p} (1+p^(1-s)-p^(-s)). - R. J. Mathar, Jan 21 2012
a(n) = Sum_{d|n} phi(d) * mu(d)^2 = Sum_{d|n} |A097945(d)|. - Enrique Pérez Herrero, Apr 23 2012
a(n) = Product_{d|n} d^moebius(n/d) (see Billal link). - Michel Marcus, Jan 06 2015
a(n) = n/( Sum_{k=1..n} (floor(k^n/n)-floor((k^n - 1)/n)) ) = e^(Sum_{k=2..n} (floor(n/k) - floor((n-1)/k))*A010051(k)*M(k)) where M(n) is the Mangoldt function. - Anthony Browne, Jun 17 2016
a(n) = n/A003557(n). - Juri-Stepan Gerasimov, Apr 07 2017
G.f.: Sum_{k>=1} phi(k)*mu(k)^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Apr 11 2017
From Antti Karttunen, Jun 18 2017: (Start)
a(1) = 1; for n > 1, a(n) = A020639(n) * a(A028234(n)).
a(n) = A019565(A087207(n)). (End)
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{primes p} (1 + p^(1-2*s) - p^(2-2*s) - p^(-s)). - Vaclav Kotesovec, Dec 18 2019
From Peter Munn, Jan 01 2020: (Start)
a(A059896(n,k)) = A059896(a(n), a(k)).
a(A003961(n)) = A003961(a(n)).
a(n^2) = a(n).
a(A225546(n)) = A019565(A267116(n)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = A065463/2. - Vaclav Kotesovec, Jun 24 2020
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)).
For n>1, Sum_{k=1..n} a(gcd(n,k))*mu(a(gcd(n,k)))*phi(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*mu(a(n/gcd(n,k)))*phi(gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = (-1)^omega(n) * Sum_{d|n} mu(d)*psi(d), where omega = A001221 and psi = A001615. - Ridouane Oudra, Aug 01 2025

More terms from several people including David W. Wilson

Definition expanded by Jonathan Sondow, Apr 26 2013

A268335 Exponentially odd numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 96, 97

Offset: 1

Vladimir Shevelev, Feb 01 2016

nonn

The sequence is formed by 1 and the numbers whose prime power factorization contains only odd exponents.
The density of the sequence is the constant given by A065463.
Except for the first term the same as A002035. - R. J. Mathar, Feb 07 2016
Also numbers k all of whose divisors are bi-unitary divisors (i.e., A286324(k) = A000005(k)). - Amiram Eldar, Dec 19 2018
The term "exponentially odd integers" was apparently coined by Cohen (1960). These numbers were also called "unitarily 2-free", or "2-skew", by Cohen (1961). - Amiram Eldar, Jan 22 2024

Peter J. C. Moses, Table of n, a(n) for n = 1..2000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74 (1960), pp. 66-80.
Eckford Cohen, Some sets of integers related to the k-free integers, Acta Sci. Math. (Szeged), Vol. 22, No. 3-4 (1961), pp. 223-233.
Vladimir Shevelev, Exponentially S-numbers, arXiv:1510.05914 [math.NT], 2015.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint arXiv:1511.03860 [math.NT], 2015.
Vladimir Shevelev, S-exponential numbers, Acta Arithmetica, Vol. 175(2016), 385-395.
D. Suryanarayana and R. Sita Rama Chandra Rao, Distribution of unitarily k-free integers, Journal of the Australian Mathematical Society, Vol. 20 , No. 2 (1975), pp. 129-141.

Cf. A002035, A209061, A138302, A197680, A000578, A000584, A001014, A001017, A008456, A010803, A010805, A010806, A010808, A010811, A010812, A001221, A124010.

Select[Range@ 100, AllTrue[Last /@ FactorInteger@ #, OddQ] &] (* Version 10, or *)
Select[Range@ 100, Times @@ Boole[OddQ /@ Last /@ FactorInteger@ #] == 1 &] (* Michael De Vlieger, Feb 02 2016 *)

isok(n)=my(f = factor(n)); for (k=1, #f~, if (!(f[k,2] % 2), return (0))); 1; \\ Michel Marcus, Feb 02 2016

from itertools import count, islice
from sympy import factorint
def A268335_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:all(e&1 for e in factorint(n).values()),count(max(startvalue,1)))
A268335_list = list(islice(A268335_gen(),20)) # Chai Wah Wu, Jun 22 2023

Sum_{a(n)<=x} 1 = C*x + O(sqrt(x)*log x*e^(c*sqrt(log x)/(log(log x))), where c = 4*sqrt(2.4/log 2) = 7.44308... and C = Product_{prime p} (1 - 1/p*(p + 1)) = 0.7044422009991... (A065463).

Sum_{n>=1} 1/a(n)^s = zeta(2*s) * Product_{p prime} (1 + 1/p^s - 1/p^(2*s)), s>1. - Amiram Eldar, Sep 26 2023

A064549 a(n) = n * Product_{primes p|n} p.

Original entry on oeis.org

1, 4, 9, 8, 25, 36, 49, 16, 27, 100, 121, 72, 169, 196, 225, 32, 289, 108, 361, 200, 441, 484, 529, 144, 125, 676, 81, 392, 841, 900, 961, 64, 1089, 1156, 1225, 216, 1369, 1444, 1521, 400, 1681, 1764, 1849, 968, 675, 2116, 2209, 288, 343, 500, 2601, 1352

Offset: 1

Henry Bottomley, Oct 16 2001

Index of first occurrence of n in A003557. - Franklin T. Adams-Watters, Jul 25 2014

			a(12) = 72 since 12 = 2^2*3 and 12*2*3 = 72.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, Vol. 113, No. 8 (2006), pp. 1732-1745; arXiv preprint, arXiv:math/0509316 [math.NT], 2005-2006.

A permutation of the powerful numbers A001694.

Cf. A003557 (a left inverse), A007947, A057521, A078310, A082695, A202535.

a064549 n = a007947 n * n  -- Reinhard Zumkeller, Jul 23 2013

[n^2/( (&+[Floor(k^n/n)-Floor((k^n - 1)/n) : k in [1..n]]) ): n in [1..50]]; // G. C. Greubel, Nov 02 2018

a:= n -> n * convert(numtheory:-factorset(n), `*`):
seq(a(n),n=1..100); # Robert Israel, Jul 25 2014

a[n_] := n * Times @@ FactorInteger[n][[All, 1]]; Array[a, 100] (* Jean-François Alcover, Feb 17 2017 *)
Table[n*Product[If[PrimeQ[d], d, 1], {d, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Jun 15 2019 *)

popf(n)= { local(f,p=1); f=factor(n); for(i=1, matsize(f)[1], p*=f[i, 1]); return(p) } { for (n=1, 1000, write("b064549.txt", n, " ", n*popf(n)) ) } \\ Harry J. Smith, Sep 18 2009

A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); factorback(f); }; \\ Antti Karttunen, Aug 30 2018

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X)/(1 - p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

Multiplicative with a(p^k)=p^(k+1) when k>0.
a(n) = n*A007947(n) = n^2/A003557(n).
Dirichlet convolution of A000027 and A202535. - R. J. Mathar, Dec 20 2011
a(n) = A078310(n) - 1. - Reinhard Zumkeller, Jul 23 2013
A003557(a(n)) = n; a(A003557(n)) = A057521(n). - Antti Karttunen, Aug 30 2018
G.f.: Sum_{k>=1} mu(k)^2*phi(k)*k*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, Nov 02 2018
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2) * zeta(s-1) * Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065463/3 = A065464*Pi^2/18 = 0.234814...
(End)
Sum_{k>=1} 1/a(k) = zeta(2)*zeta(3)/zeta(6) = A082695. - Vaclav Kotesovec, Sep 19 2020
Sum_{k>=1} (-1)^(k+1)/a(k) = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 18 2020

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

Original entry on oeis.org

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 14 2002

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

a(n) = A008683(A007947(n)).
a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010
Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011
Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011
Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011
a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013
a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018
Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023

A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

Original entry on oeis.org

4, 2, 8, 2, 4, 9, 5, 0, 5, 6, 7, 7, 0, 9, 4, 4, 4, 0, 2, 1, 8, 7, 6, 5, 7, 0, 7, 5, 8, 1, 8, 2, 3, 5, 4, 6, 1, 2, 1, 2, 9, 8, 5, 1, 3, 3, 5, 5, 9, 3, 6, 1, 4, 4, 0, 3, 1, 9, 0, 1, 3, 7, 9, 5, 3, 2, 1, 2, 3, 0, 5, 2, 1, 6, 1, 0, 8, 3, 0, 4, 4, 1, 0, 5, 3, 4, 8, 5, 1, 4, 5, 2, 4, 6, 8, 0, 6, 8, 5, 5, 4, 8, 0, 7, 5, 7

Offset: 0

N. J. A. Sloane, Nov 19 2001

Sum_{n <= x} A189021(n) ~ kx, where k is this constant. - Charles R Greathouse IV, Jan 24 2018

The probability that a number chosen at random is squarefree and coprime to another randomly chosen random (see Schroeder, 2009). - Amiram Eldar, May 23 2020, corrected Aug 04 2020

			0.428249505677094440218765707581823546...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 110.
Manfred Schroeder, Number Theory in Science and Communication, 5th edition, Springer, 2009, page 59.

R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011; Equation (116).
G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
Eric Weisstein's World of Mathematics, Carefree Couple.

Cf. A000010, A057434, A078078, A056671.
Cf. A065463, A013661.
Cf. A065473, A065480.

$MaxExtraPrecision = 800; digits = 98; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms+10]]; r[n_Integer] := LR[[n]]; (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n-1]/(n-1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

prodeulerrat(1 - (2*p-1)/p^3) \\ Amiram Eldar, Mar 12 2021

Equals A065463 divided by A013661. - R. J. Mathar, Mar 22 2011
Equals A065473 divided by A065480. - R. J. Mathar, May 02 2019
Equals (6/Pi^2)^2 * Product_{p prime} (1 + 1/(p^3 + p^2 - p - 1)) = 1.1587609... * (6/Pi^2)^2. - Amiram Eldar, May 23 2020
Equals lim_{m->oo} (1/m) * Sum_{k==1..m} (phi(k)/k)^2, where phi is the Euler totient function (A000010). - Amiram Eldar, Mar 12 2021

More digits from Vaclav Kotesovec, Dec 18 2019

A306633 Decimal expansion of zeta(2)/zeta(3).

Original entry on oeis.org

1, 3, 6, 8, 4, 3, 2, 7, 7, 7, 6, 2, 0, 2, 0, 5, 8, 7, 5, 7, 3, 6, 7, 6, 5, 8, 5, 3, 9, 8, 4, 7, 9, 1, 9, 4, 1, 1, 3, 0, 8, 1, 3, 9, 1, 4, 6, 5, 2, 4, 1, 3, 9, 2, 2, 0, 7, 7, 3, 5, 3, 1, 9, 2, 7, 6, 8, 3, 4, 4, 9, 7, 9, 7, 8, 7, 6, 0, 1, 9, 4, 2, 2, 8, 2, 2, 0

Offset: 1

Amiram Eldar, Mar 02 2019

Equals the asymptotic mean of the unitary abundancy index, lim_{n->oo} (1/n) * Sum{k=1..n} usigma(k)/k, where usigma(k) is the sum of the unitary divisors of k (A034448).
From Amiram Eldar, May 12 2023: (Start)
Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
In general, the asymptotic mean of the abundancy index of the k-free numbers (numbers that are not divisible by a k-th power other than 1) is zeta(2)/zeta(k+1) (Jakimczuk and Lalín, 2022). (End)

			1.3684327776202058757367658539847919411308139146524...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.

RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]

zeta(2)/zeta(3) \\ Michel Marcus, Mar 04 2019

Equals A013661/A002117 = 1/A253905.
Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - Amiram Eldar, Jun 23 2020
Equals Product_{p prime} (1 + 1/(p*(p+1))). - Amiram Eldar, Aug 10 2020
Equals Sum_{k>=1} mu(k)^2/(k*psi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Dedekind psi function values, A001615). - Amiram Eldar, Aug 18 2020

A066503 a(n) = n - squarefree kernel of n, A007947.

Original entry on oeis.org

0, 0, 0, 2, 0, 0, 0, 6, 6, 0, 0, 6, 0, 0, 0, 14, 0, 12, 0, 10, 0, 0, 0, 18, 20, 0, 24, 14, 0, 0, 0, 30, 0, 0, 0, 30, 0, 0, 0, 30, 0, 0, 0, 22, 30, 0, 0, 42, 42, 40, 0, 26, 0, 48, 0, 42, 0, 0, 0, 30, 0, 0, 42, 62, 0, 0, 0, 34, 0, 0, 0, 66, 0, 0, 60, 38, 0, 0

Offset: 1

Reinhard Zumkeller, Jan 04 2002

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A005117, A007947, A065463.

a[n_] := n - Times @@ FactorInteger[n][[All, 1]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 01 2021 *)

{ for (n=1, 1000, f=factor(n); k=1; for(i=1, matsize(f)[1], k*=f[i, 1]); write("b066503.txt", n, " ", n - k) ) } \\ Harry J. Smith, Feb 18 2010

a(n) = n - A007947(n).
a(A005117(n)) = 0.
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - A065463 = 0.295557... . - Amiram Eldar, Dec 05 2023

A097945 a(n) = mu(n)*phi(n) where mu(n) is the Mobius function (A008683) and phi(n) is the Euler totient function (A000010).

Original entry on oeis.org

1, -1, -2, 0, -4, 2, -6, 0, 0, 4, -10, 0, -12, 6, 8, 0, -16, 0, -18, 0, 12, 10, -22, 0, 0, 12, 0, 0, -28, -8, -30, 0, 20, 16, 24, 0, -36, 18, 24, 0, -40, -12, -42, 0, 0, 22, -46, 0, 0, 0, 32, 0, -52, 0, 40, 0, 36, 28, -58, 0, -60, 30, 0, 0, 48, -20, -66, 0, 44, -24, -70, 0, -72, 36, 0, 0, 60, -24, -78, 0, 0, 40, -82, 0

Offset: 1

Gerald McGarvey, Sep 04 2004

Also, a(n) = mu(n)*uphi(n) where mu(n) is the Mobius function (A008683) and uphi(n) is the unitary totient function (A047994), since phi(n) = uphi(n) when n is squarefree, while mu(n) = 0 when n is not squarefree. - Franklin T. Adams-Watters, May 14 2006
Conjecture: Sum_{n>=1} mu(n)/phi(n) = Sum_{n>=1} a(n)/phi(n)^2 = 0. It is true that Sum_{n>=1} mu(n)/phi(n)^s = 0 at least for s > 1 since: phi(2)=1, phi is multiplicative, so for n's that are squarefree, the phi(n) values can be partitioned in pairs where phi(m)=phi(2m) and mu(m) = -mu(2m). So Sum_{i=1..n} mu(i)/phi(i)^s < Sum_{j=floor(n/2)..n} 1/phi(j)^s, which approaches 0 as n increases since (1) n^(1-e) < phi(n) < n for any e > 0 and n > N(e) and (2) Sum_{i..n} 1/n^s converges for s > 1. Conjecture: Sum_{n>=1} mu(n)/phi(n)^z = 0 for Re(z) > 1.
Multiplicative with a(p^1) = 1-p, a(p^e) = 0, e > 1. - Mitch Harris, May 24 2005
Row sums of triangle A143153 = a signed version of the sequence such that parity = (-) iff A008683(n) = (+); 0 or (+): (1, 1, 2, 0, 4, -2, 6, 0, 0, -4, 10, 0, 12, -6, 0, 0, 0, ...). - Gary W. Adamson, Jul 27 2008
Dirichlet inverse of A003958. - R. J. Mathar, Jul 08 2011

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Wikipedia, Euler's totient function

Cf. A000010, A003958, A007947, A008683, A008966, A023900, A047994, A143153, A173557, A322581.

with(numtheory):
a:= n-> mobius(n)*phi(n):
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 06 2012

Table[ MoebiusMu[n]EulerPhi[n], {n, 85}] (* Robert G. Wilson v, Sep 06 2004 *)

a(n)=moebius(n)*eulerphi(n) \\ Charles R Greathouse IV, Feb 21 2013

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + X))[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

Dirichlet g.f.: Product_{primes p} (1-p^(1-s)+p^(-s)). - R. J. Mathar, Aug 29 2011
Sum_{d|n} abs(a(d)) = rad(n) = A007947(n). - Rémy Sigrist, Nov 05 2017
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = A065464/2 = (1/2) * Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.21412475283854722... Equivalently, c = A065463 * 3 / Pi^2. - Vaclav Kotesovec, Jun 14 2020
From Antti Karttunen, Aug 20 2021: (Start)
a(n) = mu(n)*A000010(n) = mu(n)*A003958(n) = mu(n)*A047994(n) = mu(n)*A173557(n), where mu is Möbius mu function (A008683).
a(n) = A008966(n) * A023900(n) = abs(mu(n)) * A023900(n).
a(n) = A322581(n) - A003958(n).
(End)

More terms from Robert G. Wilson v, Sep 06 2004

Edited by N. J. A. Sloane, May 20 2006

A204455 Squarefree product of all odd primes dividing n, and 1 if n is a power of 2: A099985/2.

Original entry on oeis.org

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 1, 17, 3, 19, 5, 21, 11, 23, 3, 5, 13, 3, 7, 29, 15, 31, 1, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 3, 7, 5, 51, 13, 53, 3, 55, 7, 57, 29, 59, 15, 61, 31, 21, 1, 65, 33, 67, 17, 69, 35, 71, 3

Offset: 1

Wolfdieter Lang, Jan 19 2012

There are no odd primes dividing n iff n is a power of 2.
This sequence coincides with the bisection of A007947 (even indices), which is A099985, dividing out the even prime 2 in the squarefree kernel.
a(n) divides A106609(n) for n>=1. - Alexander R. Povolotsky, Apr 06 2015

			a(5)=5 because 5 is a single odd prime.
a(9)=3 because 9=3*3 has as squarefree part 3.
a(1)=1 because 1 is a power of 2, having no odd primes as a factor.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A099985, A099984, A007947, A000265, A065463, A106609.

A204455 := proc(n)
    local p;
    numtheory[factorset](n) minus {2} ;
    mul(p,p=%) ;
end proc:
seq(A204455(n),n=1..40) ; # R. J. Mathar, Jan 25 2017

f[n_] := Select[First /@ FactorInteger@ n, PrimeQ@ # && OddQ@ # &]; Times @@@ (f /@ Range@ 120) (* Michael De Vlieger, Apr 08 2015 *)

a(n) = {my(f = factor(n)); prod(k=1, #f~, if (f[k,1] % 2, f[k,1], 1));} \\ Michel Marcus, Apr 07 2015

a(n) = factorback(setminus(factorint(n)[, 1]~, [2])) \\ Jianing Song, Aug 09 2022

a(n) = A099985(n)/2 = A007947(2*n)/2.
a(n) = A000265(A007947(n)) = A007947(A000265(n)). - Charles R Greathouse IV, Jan 19 2012
Multiplicative with a(p^e)=p for p <> 2 and a(2^e)=1. - R. J. Mathar, Jul 02 2013
a(n) = Sum_{d|n} phi(d)*mu(2d)^2. - Ridouane Oudra, Sep 02 2019
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} mu(2*n/gcd(n,k))^2.
a(n) = Sum_{k=1..n} mu(2*gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
Sum_{k=1..n} a(k) ~ c * n^2, where c = (2/5) * Product_{p prime} (1 - 1/(p*(p+1))) = (2/5) * A065463 = 0.281776... . - Amiram Eldar, Nov 19 2022
a(n) = Sum_{d divides n, d odd} mu(d)^2 * phi(d). - Peter Bala, Feb 01 2024

cp's OEIS Frontend

A065490 Exponents in expansion of constant A065463 as Product_{n>1} zeta(n)^(-a(n)).

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A007947 Largest squarefree number dividing n: the squarefree kernel of n, rad(n), radical of n.

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A268335 Exponentially odd numbers.

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A064549 a(n) = n * Product_{primes p|n} p.

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A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

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A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

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